Beyond COVID-19: Network science and sustainable exit strategies
James Bell, Ginestra Bianconi, David Butler, Jon Crowcroft, Paul C.W Davies, Chris Hicks, Hyunju Kim, Istvan Z. Kiss, Francesco Di Lauro, Carsten Maple, Ayan Paul, Mikhail Prokopenko, Philip Tee, Sara I. Walker
BBeyond COVID-19: Network science and sustainableexit strategies
J. Bell ,G. Bianconi , ,D. Butler ,J. Crowcroft ,P. C. W. Davies ,C. Hicks , H. Kim , , ,I.Z. Kiss , F. Di Lauro ,C. Maple , ,A.Paul , ,M. Prokopenko , , P. Tee and S. Walker The Alan Turing Institute, 96 Euston Rd, London NW1 2DB, United Kingdom. { jbell,chicks,dbutler } @turing.ac.uk School of Mathematical Sciences, Queen Mary University of London, London, E1 4NS,United Kingdom. [email protected] University of Cambridge, Cambridge, CB2 1TN, United [email protected] Beyond Center for Fundamental Concepts in Science, Arizona State University, Tempe,Arizona 85287 USA. { paul.davies, sara.i.walker, hkim78, ptee2 } @asu.edu ASU-SFI Center for Biosocial Complex Systems, Arizona State University and Santa FeInstitute, USA School of Earth and Space Exploration, Arizona State University, Tempe, AZ, USA Department of Mathematics, University of Sussex, Falmer, Brighton BN1 9QH, UnitedKingdom. { I.Z.Kiss, F.Di-Lauro } @sussex.ac.uk University of Warwick, Coventry CV4 7AL, United Kingdom. [email protected] DESY, Notkestraße 85, D-22607 Hamburg, Germany. [email protected] Institut f¨ur Physik, Humboldt-Universit¨at zu Berlin, D-12489 Berlin, Germany Centre for Complex Systems, The University of Sydney, [email protected] The Marie Bashir Institute for Infectious Diseases and Biosecurity, The University ofSydney, Australia.E-mail: [email protected] (Correspondence address)
Abstract.
On May 28 th and 29 th , a two day workshop was held virtually, facilitated by theBeyond Center at ASU and Moogsoft Inc. The aim was to bring together leading scientistswith an interest in Network Science and Epidemiology to attempt to inform public policy inresponse to the COVID-19 pandemic. Epidemics are at their core a process that progressesdynamically upon a network, and are a key area of study in Network Science. In the courseof the workshop a wide survey of the state of the subject was conducted. We summarizein this paper a series of perspectives of the subject, and where the authors believe fruitfulareas for future research are to be found. a r X i v : . [ phy s i c s . s o c - ph ] S e p eyond COVID-19: Network science and sustainable exit strategies
1. Introduction
Reports of a new viral infection with lethal and pandemic potential emerged in the Wuhanprovince of China in December of 2019 [1]. It was clear from early reports that this newvirus had severe respiratory complications and could have alarming fatality rates. The virus,officially designated SARS-CoV-2 by the International Committee on Taxonomy of Viruses,proceeded to create a localized epidemic in Wuhan, resulting in a severe lock down in anattempt to control the outbreak. This has subsequently proven to have not controlled theepidemic of the virus, now commonly known as COVID-19 or Corona Virus, and the worldis faced with the first widespread pandemic since the ‘Spanish Flu’ outbreak of 1918.The pathology of the disease is significantly higher than seasonal flu [2], and severe symptomscan terminate in a fatal cytokine release resulting in severe inflammatory response, high fever,hypoxia and eventually death. Initial indications from studies of the outbreak in Wuhanindicate that this fatality rate is not evenly spread demographically, with case fatality ratesamongst individuals 80 years and over being estimated between 10% to 18%, whereas ratesat half that age are barely 1% [3]. Regardless, it is clear that COVID-19 is a deadly disease,and as of July 7 th , it has claimed 551 ,
686 victims, and the initial demographics of morbidityare however likely to change as the pandemic progresses. Indeed there is strong evidence fromthe 1918 Spanish Flu that although the first wave of the infection had enhanced mortality inelderly people, the second and subsequent waves killed more indiscriminately [4] indicatingthat it is possible that the future progress of COVID-19, should there be a second wave, couldhave greater impact on the young than hitherto. In Fig. 1 we reproduce statistics from theJohns Hopkins Coronavirus Resource Center [5] for the ten countries with the largest currentoutbreaks. We note that the data on a linear scale underlines that the spread of the diseaseis accelerating, and we may be at the beginning rather than substantially into the pandemic.Public policy response to the pandemic has been a dramatic economic and societal lockdown,essentially plunging the affected nations into a kind of deep freeze. As evidence emerged thattransmission of the virus was mediated by respiratory exhalation of ‘droplets’ or, via contact,social distancing was imposed forcing millions to isolate or shelter in place. The economicimpact of this policy has been dramatic. US government data on the effect on the Chineseeconomy [6] provides an illustration of what even the short lived lock down of one regionproduced, including a 21 .
2% drop in retail sales for the whole of China, and, a drop inthe use of public transport from 70-80 million trips per day to around 10 million, as thelockdown was imposed. Around the western world the impact has begun to be felt with forexample, the United Kingdom seeing a decline in GDP of 20 .
4% in the month of April alone[7], and some commentators pointing to an economic recession that will be the most severein recorded history. eyond COVID-19: Network science and sustainable exit strategies th and 29 th , a two day virtual workshop was convenedby the Beyond Center at Arizona State University. The workshop united experts from acrossa range of disciplines, with network science being a common thread. In the paper we seekto summarize the perspectives that were shared at this workshop, and point to where theparticipants felt the most promising areas of study currently lie. As such, this paper hopesto provide a perspective upon the state of the understanding of this extremely important eyond COVID-19: Network science and sustainable exit strategies This paper is a curation of summary reports authored by the speakers at the workshop. Inorder to provide a coherent overview of the discussion we have organized the paper into threemajor parts. We begin with an overview of the theoretical network models in Sections 2 and3. Here we cover the latest techniques that are being used to analyze how the standardepidemic models of such as SIR/SIS are enhanced by the considerations of the structure ofcontact networks. It can be shown that this has a dramatic effect on the disease inducedherd immunity (DIHI) threshold, and has important consequences for the management ofthe pandemic. Specifically the structural nature of the contact network and the details ofany actions such as lockdowns impacts significantly the procession of the disease.In Section 4 we discusses novel theoretical questions raised by the modeling of COVID-19. We will reveal the role of stochasticity and criticality in plateauing time series, whichprovides an important additional source of uncertainties too often neglected in modelingframeworks. Moreover we indicate that network science is the most suitable tool totheoretically investigate the efficiency of the track and tracing apps at the system level.In fact the network science approach allows to capture non-linear effects of the spreadingdynamics.In Section 5 we turn our attention to another important modeling technique, Agent BasedModeling (ABM), which has been successfully applied to the initial progress of the epidemicin Australia. This approach permits the detailed use of scenario planning to model, forexample, the effects of social distancing measures. By essentially creating a digital ‘twin’ ofthe epidemic this technique accurately predicted many of the features of the early progressof the disease, incorporating many complex features of the demographics and geography ofthe country.Key to the management of the epidemic has been the use of so-called ‘track and trace’,whereby the contacts of infected individuals are identified and isolated to prevent furtherinfection. In Section 6 we give an overview of the technology of contact tracing applications.In this section, we discuss the requirements for identity systems to support these apps, andthe way that there are common design patterns for how the apps and the identity systemsthemselves can be implemented. This has important consequences for various properties ofthe systems in terms of protection of privacy with regards to identity, and attributes (“isinfected”) associated with that identity.In Section 7 a theoretical analysis of the effectives of contact tracing approaches toepidemic control is explored. Building upon a probabilistic model the conclusion is thatparticipation in such a control approach needs to be widespread, although the precise degree eyond COVID-19: Network science and sustainable exit strategies
2. Network Science and Epidemiology
The quantitative analysis of epidemiology dates back to the work of Kermack andMcKendrick [12, 13, 14], almost a century ago. Their work evolved into the familiar“compartmental” models, which explore the dynamics of a population whose individualsmove between the “compartments” or categories of infections. There are a number of variantsof such models that use different compartments, and different schema for the transition of anindividual from between them. As an illustration we will briefly describe one of the betterknown variants, the SIR model. Here the compartments are characterized by the time-seriesof the numbers of individuals in the population that are susceptible to infection (S), areinfected (I) and have recovered from the disease or have died (R). The time evolution ofthe population in each of the compartments is therefore time-dependent, and encodes thedynamics of the epidemic. The evolution of the epidemics may be described by a system ofOrdinary Differential Equations (ODEs) that can be solved to determine the progress of thedisease.A central, and very important, assumption of the SIR model is that the population mixeshomogeneously. This means that the probability that a randomly chosen member of thepopulation is infected (moves from S → R ) is uniform, which greatly simplifies the analysis.Furthermore, an individual in the susceptible population can encounter and be infected byany member of the infected population, and the only factors that impact the dynamics arethe relative sizes of these populations. This explicitly excludes finer details such as locationfactors, where in a particular area susceptible individuals may be present at lower densitycompared to the average.For a population of total size N , the system may be analyzed with three coupled ODEs[15, 9] eyond COVID-19: Network science and sustainable exit strategies dS ( t ) dt = − βI ( t ) S ( t ) N , (1) dI ( t ) dt = βI ( t ) S ( t ) N − γI ( t ), (2) dR ( t ) dt = γI ( t ). (3)Here S ( t ), I ( t ),and R ( t ) represent the time varying numbers of the population in eachcompartment, β represents the rate a susceptible person is infected and γ the rate at whichan infected individual either recovers or dies. There will be βI ( t ) infecting encounters in unittime with the S ( t ) /N portion of the population available to be infected, and a number γI ( t )of infected individuals either recover and become immune to infection or die. In Section 3these equations are treated in more depth, and the reader should note that there the symbol τ is used in place of β . The ratio of the two parameters R = βγ , is known as the basicreproductive ratio, and it is widely known by the general public as the key measure of theseverity of the spreading of the disease. A simple interpretation of R is the average numberof susceptible persons that are infected by an infected individual. Therefore large valuesof R are associated with public policy responses such as lockdowns, and the ubiquitousreferences to exponential growth and doubling times. Indeed by inspection of Eqs. (1) wecan see that for R >
1, that I ( t ) increases with time, and conversely for R < R > supercritical regime and the epidemics outbreak will affecta finite fraction of the population. For R < subcritical regime andthe epidemics outbreak dies out affecting only an infinitesimal fraction of the population. For R = 1 the epidemics is in the critical regime and is the epidemics starts with a single infectedindividual it will infect an infinitesimal fraction of the population however the epidemics willbe affected strongly by stochastic effect and the size of the outbreak is difficult to predictand can have very wide fluctuations.The simplistic SIR model is problematic for many reasons, not least of which is that it is notpossible in general to assess the value of R at a given point in time; it is generally inferredretrospectively using historical data. Indeed the model assumes that R as defined as theratio of β to γ is fixed in time, and of course as measures are taken to combat an epidemicthis is not generally true. Instead we can refer to R as a point in time reproduction rate,which is possible to determine in many ways using historical data. One such method is touse macroscopic parameters such as the doubling time and the average duration t c betweenan individual becoming infected and going on to infect another person. This requires certainassumptions regarding the distribution of macroscopic parameters, and to an extent theunderlying epidemic model, but it allows for an empirical estimation of the value of R [16].For example, if we define r as the growth rate of an infection in a unit time (number of newcases per day for example), in the most naive linear model R = 1 + rt c , which relies uponthe assumption that t c is constant. Alternatively is we apply a simple multiplicative growth eyond COVID-19: Network science and sustainable exit strategies R = exp ( rt c ). (4)The values of R obtained using these two different models can be very different. Thishopefully serves to illustrate that the actual value of R is very much dependent uponassumptions implicit in the underlying model. In short its use is problematic in isolation asa guide to the severity and progress of an epidemic. Infections require a physical mechanism of transmission between infected and susceptibleindividuals. This does not have to be physical contact (as in the case of sexually transmitteddiseases), but can arise when individuals are within close enough proximity for sufficient timethat an airborne pathogen can pass from one individual to another. The precise biologicalmechanism for the transmission of the disease is assumed to be encoded in the parametersof model operating on these networks.One is naturally led to the concept of contact networks, where one represents individuals(or places) as nodes, and the links represent individuals coming into contact. Startingwith a number of infected nodes the epidemic then proceeds by transmission via the linksaccording to a certain transmission probability. Clearly the structure of this network canhave a profound effect upon the progress of the disease, as the degree of a node (number oflinks) represents the number of people available to be infected by that individual. When thephrase breaking the chain of infection is used, this translates into isolating infected nodesby severing these links, being the basis of quarantine measures.Models of real world networks are extremely well studied, and it has been known for sometime that many of them possess the “small world” property [17], whereby non-locality inthe connectivity of nodes in the network creates shortcuts in the network, which has animportant effect on the propagation of an epidemic.An important property of real world networks, such as contact graphs, is their degreedistribution P ( k ) of nodes, which quantifies the probability of a randomly selected nodehaving k links. Real world networks are often scale-free, i.e. they have a power-law degreedistribution of the form P ( k ) ∝ k − α , with values of α typically in the range 2 . < α < . α (cid:39) eyond COVID-19: Network science and sustainable exit strategies degree class approximation of Pastor-Satorras and Vespignani [24].This approach assumes that all nodes of a given degree are statistically equivalent, andapplies a variant of the SI model (nodes are designated to be either susceptible or infected)to each cohort of nodes sharing the same degree. It is possible to solve these equationsusing an underlying assumption of the degree distribution, and obtain an expression for thecharacteristic time for the spread of the infection τ SI in terms of the moments of k : τ SI = (cid:104) k (cid:105) β ( (cid:104) k (cid:105) − (cid:104) k (cid:105) ) . (5)The value τ SI enters into the dynamics as a time-scale factor. If i k is the fraction of nodesof degree k that are infected, it can be shown that di k /dt ∝ ke t/τ SI . As (cid:104) k (cid:105) → ∞ for scalefree graphs this has the very surprising consequence that the characteristic spreading timefor the disease is zero. In essence, the disease propagates extremely fast affecting rapidly alarge fraction of the network, because of the presence of hubs in the network, which allowan infected individual to rapidly infect a large fraction of network of contacts.The SIR model can be mapped to link percolation which can then be used to predict theexpected size of an outbreak as a function of the trasmissibility of the disease T [25, 8]. Inthis approach the nodes of the network represent single individuals and the links representthe social contacts of an individuals providing the possible routes for the transmission ofthe disease. The transmission of the diseases from an infected to a neighbour susceptibleindividual occurs with a probability T called ’transmissibility’, which is dependant upon thelength of time an individual is infectious τ , and the rate β at which an individual infects oneof its contact. Assuming that the rate β is independent on time, the transmissibility can beexpressed as T = 1 − e − βτ . (6)Therefore the SIR outbreak can be represented as a network in which the route oftransmission is indicated by the “occupied’ links’, where each link is occupied with probability T . Therefore the sub-graph connected only the occupied links, indicates the infected clusterof individuals. As a function of T this cluster can be very small, affecting an infinitesimalfraction of all the node of the networks or very large, or giant if it includes a finite fraction eyond COVID-19: Network science and sustainable exit strategies T . For T smaller than the critical value T c the largest connected cluster involves only an infinitesimal fraction of the nodes of thenetwork, implying that for these value of the transmissibility the epidemic dies out beforebecoming widely spread in the population. However for T larger than the critical value T c a GC emerges indicating that the outbreak affects a finite fraction of the population (heremodelled by its corresponding contact network).Very interestingly there is an important effect of the degree distribution on the criticalproperties of the SIR model. The critical value T c of the transmissibility , called the epidemicthreshold , on a random network with given degree distribution is given by [22, 23] T c = (cid:104) k (cid:105)(cid:104) k (cid:105) − (cid:104) k (cid:105) . (7)This result implies that if the network has a homogeneous degree distribution, with a finitesecond moment of the degree distribution (i.e. with (cid:104) k (cid:105) < ∞ ) the epidemic threshold T c is finite. Therefore if the transmissibility T is smaller that the epidemic threshold T c the outbreaks does not affects significantly the population. However for scale-free networkshaving diverging second moment of the degree distribution (i.e. with (cid:104) k (cid:105) → ∞ ) the epidemicthreshold vanishes T c → N of the nodes of the network diverge, i.e. N → ∞ .This means that also epidemics with very small transmissibility can become pandemics.This result implies that globalized societies are very prone to pandemics as the air-travelconnections are well known to be described by scale-free networks.An important network property of disease spreading is the fact that the nodes of the networkare not equally likely to get the infection. In particular nodes of high degree are more likelyto get the infection of nodes with lower degree. Therefore it comes as no surprise that peoplewith many contacts including politicians and bus drivers have been more likely to be infectedin the first wave of COVID-19.This result is rather intuitive as if we assume that each connection of an individual is equallylikely to be route for transmission of the disease, an individual with more connections willbe more likely to get the disease than an individual with less connections. The Herd Immunity (HI) threshold is thepercentage of the population which, if immune, prevents the number of infected individualsfrom growing due to the scarcity of susceptible individuals. A classic result from non-network eyond COVID-19: Network science and sustainable exit strategies R as h = 1 − R . (8)This is straightforward to derive by noting that for compartmental models a non-growingepidemic requires that R (1 − h ) = 1. With an estimate of R for COVID-19 of between2 . − .
0, we arrive at a value of h between 50% and 66% needed for the epidemic to peakin absence of containment measures. This is a very important value as the lower it is, themore likely that the epidemic will naturally abate, but for reference at the time of writingonly 2% of US population is currently infected.In addition to HI, there is a related concept Disease Induced Herd Immunity (DIHI) thatcan occur as the natural result of the first wave of infection, and is not dependent uponintervention measures. In essence this relies upon the disease spreading quickly through themore highly connected nodes early in the epidemic, such that if any intervention such aslockdown is eased the network of contacts is sufficiently disrupted to prevent further spread.Using the mapping of the SIR model to percolation, it is rather intuitive that the removalof links or nodes from the contact graph can reduce the GC through which the epidemicpreferentially progresses, and would at a certain point prevent the growth of the epidemic.This removal of links or nodes is represented in the public health domain by immunizationor social distancing measures. In fact immunized individuals or individuals in quarantineare nodes of the network that cannot spread the epidemics any more. Critically, for scale-free networks, the behavior of the size of the GC as nodes are removed is different fromother random networks [28, 26]. Targeted measures that remove nodes of high degree wouldcollapse the GC more quickly than would random removal (by immunization or isolation).The foregoing important insight is readily understandable in terms of the so-called friendshipparadox, first introduced by Scott Feld [29]. It follows from the simple observation that onaverage your friends have more friends that you do. Because your friendly friends havemore contacts they are more likely to be immunized by targeted immunization, leading toan improved efficiency of targeted immunization than of random immunization in scale-freenetworks.Note however that the efficiency of this targeted immunization on a scale-free networkstrongly depends on two parameters, the power-law exponent α of the power-law degreedistribution and the minimum degree of the nodes. Let us assume T = 1 let us consider ascale-free network with degree distribution P ( k ) = Ck − α with m ≤ k ≤ α = 2 . m = 2 in order to suppress the epidemics it is necessary to perform atargeted immunization over a fraction of a population given by f (cid:39)
20% but this fractionincreases to f (cid:39)
45% for m = 4 and f (cid:39)
59% for m = 6. Therefore the efficiency ofthe targeted immunization strategy is very sensitive to the minimum degree in the network eyond COVID-19: Network science and sustainable exit strategies
3. Overview of Network Based Epidemic Models
The transmission of a disease depends not only on the intrinsic characteristics of the pathogenthat causes it, but, equally importantly, on the network of disease-transmitting contactswithin the population. If this contact structure is ignored and homogeneous random mixingis assumed then it is well known that if a fraction 1 − / R of the population cannotbe infected (e.g. vaccinated preventively or already infected) then the residual susceptiblepopulation can no longer sustain an epidemic. Instead, a recent observation [31] is that, bytaking into account heterogeneities in the population (to be understood in the broadestsense but here captured as heterogeneities in contacts), this threshold can be crossedwhen far fewer individuals have been infected. This is because the disease acts like atargeted vaccine, preferentially ‘immunizing’ higher-risk individuals who play a greater rolein transmission. Therefore, a controlled ‘first wave’ may leave behind a residual populationthat can no longer sustain a ‘second wave’ once interventions are lifted. This concept of eyond COVID-19: Network science and sustainable exit strategies The exact ODE system for the SIR model describes the evolution for the expected numberof nodes in given statuses. A formal derivation from the exact system is given in [9]. Thesystem of equation is:˙[ S ] = − τ [ SI ] , ˙[ I ] = τ [ SI ] − γ [ I ] , ˙[ R ] = γ [ I ] , (9)where [ SI ] is the expected number of links between susceptible and infected nodes. Thissystem is not closed: to solve it we need an expression for the evolution of the expectednumber of S − I links (expected number of pairs), which in turn requires us to describe thesystem at the level of triples, and so on. Mean-field models curtail this expansion at somelevel, by expressing higher order quantities in terms of lower order ones. These methods arealso known as closures since they lead to a self-consistent system of differential equations.Such closure are often found by taking into account the network structure up to a certainlevel (e.g. mean-degree, degree distribution and clustering for example). eyond COVID-19: Network science and sustainable exit strategies The simplest mean-field model incorporates solely the average degree (cid:104) k (cid:105) of the network,and it can be described by the following equations˙[ S ] = − τ (cid:104) k (cid:105) N [ S ][ I ] , ˙[ I ] = τ (cid:104) k (cid:105) N [ S ][ I ] − γ [ I ] , ˙[ R ] = γ [ I ] . (10)The intuition behind this closure is to consider all the nodes as having the same degree (cid:104) k (cid:105) .There are [ S ] susceptible nodes with (cid:104) k (cid:105) [ S ] stubs connecting them to their neighbors. Weassume that infected nodes are distributed randomly on the network, so that the probabilitythat a neighbor is infected is [ I ] N . The expression for [ SI ] is then given by[ SI ] ∼ (cid:104) k (cid:105) [ S ] [ I ] N .
In the homogeneous mean-field model (10), the system is closed by two approximations:each node has the same degree and infected nodes are uniformly distributed on the network.The degree-based mean-field model (also called Heterogeneous Mean Field [33]) improves theclosure by removing the first approximation, i.e. by incorporating the degree distribution inthe system. We denote with [ S ] k ( t ) the expected number of susceptible nodes with degree k at time t , similarly for [ I ] k and [ R ] k . We define [ S ] = (cid:80) ∞ k =1 [ S ] k , similarly [ I ] and [ R ]. As inthe homogeneous mean-field model, we average the infection pressure across all the infectednodes. The resulting ODEs are˙[ S k ] = − τ k [ S k ] π I , ˙[ I k ] = τ k [ S k ] π I − γ [ I k ] , ˙[ R k ] = γ [ I k ] ,π I = (cid:80) M(cid:96) =1 (cid:96) [ I (cid:96) ] (cid:80) M(cid:96) =1 (cid:96)N (cid:96) , (11)where N (cid:96) = P n,p ( (cid:96) ) N is the number of nodes with degree (cid:96) , and P n,p ( l ) is the negativebinomial or versions of it used later to simulate degree distributions with differentheterogeneities. This system keeps track of the degree distribution and hence theheterogeneity in it, but mixing between nodes of different degrees happens at random butproportionally to degree [33, 9]. In reality, correlations build up: if a node has a neighborinfected, it is more likely to be infected itself than if it were randomly selected among thesusceptible nodes. Hence, better models are required, which results in closures at the levelof triples. eyond COVID-19: Network science and sustainable exit strategies In the heterogeneous pairwise model, we also consider the expected number of linksconnecting a node of degree k in state A to a node of degree (cid:96) in state B [36, 9], thatis [ A k B (cid:96) ]. To include these quantities into anode system, we need to write an expressionfor triples of the form [ A k B (cid:96) C m ]. By doing so, we are effectively including the correlationsbetween nodes in the same state. The closure is done at the level of triples (i.e. triples areapproximated by singles and pairs), and hence an approximation for the triples are needed.In this framework we also consider clustering (i.e. the propensity with which two neighborsof a node are themselves connected). The closure in this case is:[ A k B (cid:96) C m ] = (cid:96) − (cid:96) (cid:18) (1 − ϕ ) [ A k B (cid:96) ][ B (cid:96) C m ][ B j ] + ϕ [ A k B (cid:96) ][ B (cid:96) C m ][ C m A k ][ A k ][ B (cid:96) ][ C m ] (cid:19) , (12)where ϕ is the global clustering coefficient in the network. For the un-clustered case wesimply set ϕ = 0. The derivation of this can be found in [36], for example. The resultingODEs are, ˙[ S k ] = − τ (cid:88) (cid:96) [ S k I (cid:96) ] , ˙[ I k ] = τ (cid:88) (cid:96) [ S k I (cid:96) ] − γ [ I k ] , ˙[ R k ] = γ [ I k ] , ˙[ S k I (cid:96) ] = − γ [ S k I (cid:96) ] + τ (cid:32)(cid:88) α [ S k S (cid:96) I α ] − (cid:88) α [ I α S k I (cid:96) ] − [ S k I (cid:96) ] (cid:33) , ˙[ S k S (cid:96) ] = − τ ([ S k S (cid:96) I ] + [ IS k S (cid:96) ]) , ˙[ I k I (cid:96) ] = − γ [ I k I (cid:96) ] + τ (cid:32)(cid:88) α [ I α S k I (cid:96) ] + (cid:88) α [ I k S (cid:96) I α ] + [ S k I (cid:96) ] + [ I k S (cid:96) ] (cid:33) , (13)where triples are closed using equation (12). This system includes t both the full degreedistribution of the network and the evolution of the [ SI ] pairs. The number of equationsin the heterogeneous pairwise model grows very large if the network has degrees of manydifferent types (since there is an equation for ˙[ S k I (cid:96) ] for every k , (cid:96) pair). The main focus here is to investigate the impact of degree-heterogeneity and clustering onherd immunity induced by the first wave of the epidemic, also known as disease induced herdimmunity (DIHI) [31]. In networks with heterogeneous degrees, the epidemic typically findsthe high-risk groups first and thus ‘removes’ important individuals or risk groups. In linewith [31, 37], we exploit this fact and consider different levels of degree-heterogeneity usingthe degree-based mean-field and heterogeneous pairwise models to explore what happens in eyond COVID-19: Network science and sustainable exit strategies (cid:104) k (cid:105) = 10 in allthe panels, where the variace is tuned by the parameter p in (14) to be, from left to right, σ = 1, σ = 30 and σ = 300, respectively.the wake of a lockdown period when some level of spreading is possible. For illustrativepurposes, we set the degree distribution of the network to be a negative binomial of the form P n,p ( k ) = (cid:18) k + n − n − (cid:19) p n (1 − p ) k . (14)The reason for this choice is that we want to highlight how heterogeneities in the contactstructure play a central role in determining the DIHI. To illustrate this point, we typicallyconsider three different degree distributions of increasing heterogeneity. An example of thisis shown in Fig. 2 with (cid:104) k (cid:105) = n (1 − p ) /p = 10, while the variance is tuned using the secondfree parameter.We consider a SIR outbreak on a fixed population of size N = 6 . · (in line with manywestern countries, such as the UK). We arbitrarily set the recovery rate γ = and the per-contact rate of infection and average degree are given in the figure captions. We initializethe outbreak by infecting I = 5 nodes at random in one of the compartments and let theepidemic run until 0 .
5% of the population gets infected. Then, a lockdown policy of duration T reduces τ → ˜ τ = ατ . Afterwards, lockdown is lifted and τ returns immediately to itspre-lockdown value.Fig. 3 shows results based on the heterogeneous pairwise model without clustering fornetworks with increasing levels of degree heterogeneity (from left to right). Similar results(not shown) hold for the degree-based mean-field models [38], with generally higher epidemicsin the latter. In each case, we find the optimal α (a simple down scaling of the transmissionrate without change to the network) and report the number of infections required to achieveDIHI (i.e. total number of infected and recovered nodes at the end of lockdown such that the eyond COVID-19: Network science and sustainable exit strategies α (see legend) and DIHI in delta-like (left), normal-like (centre) and scale-free-like (right) networks using the heterogeneous pairwise model with ϕ = 0. Continuouscurves indicate [ I ]( t ), while dashed curves indicate [ R ]( t ). The two vertical curves representthe beginning and the end of the lockdown. Horizontal lines and the correspondingpercentages are the cumulative prevalence at the end of lockdown for the optimal strategy.Here, (cid:104) k (cid:105) = 10 and τ = 0 . ϕ = 0 .
5. Vertical lines areat the beginning (continuous) and end (dashed) of control. The blue continuous curve isoptimal control for ϕ = 0, the dashed brown is optimal control for ϕ = 0 .
5. For comparison,the continuous brown is the optimal control for ϕ = 0 when applied to a network with ϕ = 0 .
5, the dashed blue line vice-versa. (right) Impact of variance in degree distribution onDIHI h d , for different pairwise models with different values of ϕ (see the legend). Controlduration is 100 days from the moment I ( t ) + R ( t ) ≥ . τ = 0 .
04. The variance of the degree distribution used for the left panel is 15,corresponding to the second point on the x-axis in the right panel. eyond COVID-19: Network science and sustainable exit strategies α ) leads always to a sustainedsecond wave. Conversely, if lockdown is not strict enough (high value of α ) the epidemicwill run its course during the first wave with some reduction in the final size. Hence, thereis an optimal value of α for which the final epidemic size is smallest and the epidemic post-lockdown is subcritical.If we include a clustering coefficient bigger than 0 in the analysis, Fig. 4, we generally observelonger durations of the epidemic and smaller peak prevalence, compared to the unclusteredcase (see also [39]), with an overall smaller final size. This suggests that, all else beingequal, it is possible to achieve the same herd immunity level with less aggressive lockdownmeasures, if the network is clustered. It is worth noting that the final epidemic size is alsosmallest at the optimal α value (see also [31]).Opting for the more accurate heterogeneous pairwise model, the level of DIHI is plotted forincreasing values of variance in the degree distribution and for different clustering levels, seeFig. 4. It is clear that higher variance can drive DIHI levels to as low as 30%. The impact ofclustering tends to lower the DIHI, but its effect is negated if the variance is high. This showsthe non-trivial interactions between network properties where clustering has biggest impactin sparse networks and where high levels of degree heterogeneity can washout the effectof clustering. Furthermore, the same plot shows the DIHI levels based on the degree-basedmean-field model. The trend is in line with results based on the heterogeneous pairwise modelalbeit with somewhat higher DIHI levels. However, the DIHI-levels are is sharp contrast,that is being much smaller, compared to 1 − / R (cid:39) .
76 ( R = (cid:104) k (cid:105) τ /γ (cid:39) . eyond COVID-19: Network science and sustainable exit strategies
4. Criticality and Network Effects on Epidemic Containment Measures
The theoretical interpretation of the data on the COVID-19 epidemics has proven to bevery challenging. The data quality, the testing policies and the methodology to recordfatalities varies widely among different countries. These effects are very significant andallow a true comparison of the time series of infected individuals only within a country.Among different countries, despite the fact that comparisons based on infected individualsare done routinely by news outlets, only the comparison based on excess deaths data seemto provide an unbiased measure of the global impact of the epidemics in the society. Alsoif we neglect the challenges connected with the data quality modelling COVID-19 needto face important other factors. As the first pandemics in a global and connected humansociety, different factors play their role in determining the efficiency of forecasting algorithmsincluding containment measures, adaptive behavior of the populations and opinion dynamics.Consequently, for scientists working on epidemic spreading, predicting the evolution of thepandemic is a continuous effort of including data about human contacts and behavior intothe model, inform the governments and the population, and then adapt again the modelto the novel adaptive response of the population resulting in a “weather forecast” of theepidemic spreading. This type of research is quite suitable for Agent-Based-Models andNetwork Science models [40] at the meta-population level which include a compartmentaldescription of the society and the mobility of the populations across different spatial regions.Another challenge posed by the COVID-19 pandemics is that COVID-19 is a airbornedisease. This implies that the spreading routes are strongly affected by spatial proximity eyond COVID-19: Network science and sustainable exit strategies
At the onset of the COVID-19 pandemics the time series of infected individuals and deathsclearly followed exponential growth at each epidemic focus. The doubling time of theseexponentials ranged between two and three days in Europe at the onset of the epidemicsproviding evidence for a similar spreading dynamics well captured by the SIR dynamicsin well-mixed populations. To ”flatten the curve” of the number of infected individualstwo types of containment measures were adopted. The first one, includes the lock down andfocuses on reducing the number of contacts of each individuals. The second one implies a fastdetection of cases and includes efficient track and tracing strategies [42, 43]. After the firststages of the evolution of the epidemics, when containment measures have been implemented,epidemic time series started to show characteristic plateaux not typically encountered inepidemic models [43]. The typical SIR evolution of an epidemics with constant infectivity R includes an exponential onset of the number of infected individuals, and an epidemic peakmarking the characteristic time at which the infection has spread to a large fraction of thepopulation, producing herd immunity and causing the reduction of the number of infectedindividuals in time. This is the scenario expected in the SIR dynamics when we are in the socalled supecritical regime with infectivity R >
1. In this case the epidemic spreading resultin the infection of a finite fraction of the population, leading to many fatalities if containmentmeasures are not put in place to control the spread of the virus. A particular feature of theSIR time series in this supercritical regime is that the peak is well defined and not plateauingas long as R >
1. In Ref. [44] an explicit calculation shows that plateauing time-series are eyond COVID-19: Network science and sustainable exit strategies critical with R (cid:39) n of infected individuals is greater than one, i.e. n >
1. This critical SIRdynamics is characterized by a power-law growth of the number of removed individualsreported in several countries at the later stage of the epidemics [47, 48] and is stronglyaffected by fluctuations, which make predictions of the duration of the outbreak and theirsize very challenging. For this reason in this regime it is crucial to abandon deterministicmodelling of epidemics and embrace a full stochastic modelling of the epidemic spread [44].In Fig. 5 we show two examples of stochastic time series of the SIR critical dynamics showingthe important effect of stochasticity and providing evidence that plateauing time series canspontaneously occur for critical SIR dynamics with non-trivial initial condition.In order to describe the observed COVID-19 plateauing time series many modellers presentlyconsider adaptive models that consist in increasing and decreasing the infectivity R in time.It is possible that human adaptive behavior can be modelled in this way, however this is nota necessary assumption to obtain plateauing time series. Moreover most of these ad hocmodels might still strongly underestimating the role of fluctuations as long as they rely ondeterministic models. Determining what is the fraction of adoption of tracing apps that would guarantee a goodefficiency of the technology and the control of the epidemic spreading is a fundamentalproblem in COVID-19 research activity.Despite the problem is a inherently network problem most of the attempts to address thisproblem rely exclusively on linear dynamics [50, 51].Network theory can provide an important contribution by allowing to capture non-lineareffects of the epidemic spreading thanks to the mapping of this process to percolation[52, 53, 54].In a recent paper [49] the mathematical framework to fully capture the role that automatedtracing app have on epidemic spreading has been proposed. In this model everynode/individual of the network is assigned a variable indicating whether the node has adoptedor not the app. Infected individuals transmit the disease to a susceptible neighbours withprobability T , called the transmissibility of the epidemic unless they have the app and theyhave been infected by individuals with the app (see Fig. 6). This theoretical model fullycapture all the non-linear nature of the spreading process and can be solved on real networks eyond COVID-19: Network science and sustainable exit strategies Time I n f ec t e d Time R e m o ve d Time I n f ec t e d R e m o ve d (b)(a)(c) (d) Figure 5: Epidemic time series generated by the stochastic SIR model at criticality startingfrom non-trivial initial conditions. Panel (a,c) show two time series of the number of infectedindividuals while panels (b,d) show the corresponding time series of removed individuals.All time series correspond to a population of N = 10 individuals and an initial number ofinfected individuals given by n = 128.Despite the panels (a,b) and (c,d) are generated usingthe same model with the same parameters the resulting two SIR dynamics are significantlydifferent. In particular the outbreak size and the outbreak duration are very different in thetwo simulations due to stochastic effects.using message passing techniques. This model indicates that also a moderate adoption ofthe tracing app can have a significant impact in slowing down the spread of the epidemics[49].
5. Agent-based modeling
Stochastic agent-based models (ABM) have been successfully used for modeling the COVID-19 pandemic, evaluating non-pharmaceutical intervention strategies, as well as providingtimely policy advice [55, 56]. For example, the ABM approach to tracing and controlling eyond COVID-19: Network science and sustainable exit strategies T called the transmissibility unless the infected individual has the app andhas been infected by another individual with the app as discussed in Ref. [49].the pandemic in Australia [56] compared several mitigation and suppression strategies,and pinpointed an actionable transition across the levels of social distancing compliance,in the range between 70% and 80% levels. Specifically, a compliance at any level below70% was shown to be insufficient to reduce incidence and prevalence, for any duration ofsocial distancing, even when coupled with effective mitigation (i.e., case isolation, homequarantine), and international travel restrictions. In contrast, under the same mitigation andborder control conditions, a compliance at the 90% level was found to control the diseasewithin 13–14 weeks. In addition, this ABM accurately predicted key features of the firstwave in Australia: the peaks of incidence and prevalence in late March and early April 2020respectively, and the range of cumulative incidence attained after the suppression period atthe end of June 2020. It also identified formation of the second wave in early July 2020,once the suppression strategy is relaxed [56] .This predictive accuracy was achieved by utilizing a high-resolution individual-basedcomputational model calibrated to both demographic features of the Australian population(based on the Australian census data) and key characteristics of the COVID-19 pandemic.The demographic component was validated previously, in context of pandemic influenzamodeling [57, 58, 59, 60], while the COVID-19 epidemiological component was validated in anow-casting mode in March 2020. Furthermore, the model was cross-validated by a genomicanalysis of COVID-19 activity in New South Wales (NSW), the most populous state ofAustralia [61], focused on locally acquired clusters in the state. In particular, the fractionsof local transmissions inferred by the ABM were compared against the genomic sequencingof SARS-CoV-2, carried out during February–March 2020 in a subpopulation of infected eyond COVID-19: Network science and sustainable exit strategies eyond COVID-19: Network science and sustainable exit strategies
6. Overview of Contact Tracing Apps
Identity services [80] are used to uniquely distinguish a specific person from others (whoare you?). This allows the association of that unique person with attributes, establishing eyond COVID-19: Network science and sustainable exit strategies
Decentralized Identity Systems [81] and decentralized contact tracing [82].
Eachperson creates and curates their own data. There is no external service (deemed selfsovereign). If someone wants to found out who I am and what I can do, they ask me.Data still needs to be somehow ratified by some authority in the first instance, but fromthen on you are your own authority. You can vouch for yourself.
Centralized Identity Systems [83] and centralized contact tracing [84].
Everyoneplaces his or her data in a server; colloquially sometimes referred to as putting all youreggs in one basket. A weakness in this is that if you drop the basket, or someone takesthe basket, you have no eggs.For centralized systems, not just for identity services, in general we usually combine anumber of technologies to provide assurances against certain problems such as the loss ofconfidentiality. For example, services such as Authentication, Access control, Authorizationsupport, and we hope encrypt the data at rest, in transit, and during processing. Modernsystems can also run servers in Secure Enclaves, e.g. using Intel’s SGX or ARM’s Trustzoneor similar, which provide for hardware enhancements to improve resistance against attacks onprivacy. One can also employ software techniques such as Fully Homomorphic Encryption(FHE) to carry out the lookups, without decrypting the query or response or data beingqueried. In addition, Differential Privacy techniques permit you to control how much isrevealed by a query to a central system. in some cases, for example when the querier wantsto learn aggregate statistics, access is permitted, but is denied for a specific identity/attributepair which can reveal personal data, (e.g. “this person is over 21”). eyond COVID-19: Network science and sustainable exit strategies
Symptom reporting
No need for id, but needs uniqueness of reports - inherently it isabout medical stats at a given time ...token are more than good enough. There’s neverany need to re-link a report to an individual.
Contact tracing
Centralized and may require id, as the reason for centralized systemsmay be to combine health status with other factors (age, gender, ethnicity) forepidemiological studies and then relink to other ids to see if if there is any variation ininfections between different groups. Indeed, one can also uncover immunity expiry (e.g.via re-infection, if and when that occurs). eyond COVID-19: Network science and sustainable exit strategies
Immunity passporting
May need id, if required e.g. as a visa to accompany a realpassport to allow for travel: can be centralized, or could be decentralized where eachuser holds immunity status and just has to show it associated with a foundational id toverify.
Threats exist to the correct and trustworthy operations of Identity Systems (and apps). Whatare the threats, and from whom do they originate? This is not an exhaustive list, but toillustrate the range of considerations that might impact on the cost of various implementationpatterns for id or a health app.
The Human level • Fool the system (e.g that I am under 21, or I am immune). • Fool or coerce people to register/deregister ( commonly known as a masquerade),or require immunity passport to return to work, e.g. avoiding cost of providing asafe workplace.. • Fool or coerce people to verify credentials on behalf of someone, i.e. spoofing. • Run spoof service, so people give biometric, and other info, to a fake interface.
The Technology level • Does the system actually provide minimal answers (e.g. “isover 21 not is 24)? Can the user have confidence that the system only displays ananodyne yes or no, and only the client knows what was queried (for example “isyour age over 21)? • Exploit vulnerabilities in service to do the above individually • Id theft, masquerade, prevention of service, etc • DDoS service across a wide range of servers... • Indirect attacks (e.g. on network & power infrastructures)
The Organizational level (insider attacks, state actors etc) • Surveillance of use(and meta data use e.g. what id is used when & where) eyond COVID-19: Network science and sustainable exit strategies • Surveillance of register/deregister (set membership attacks) • Isolation of subgroups by attribute, for differential treatment...What would make ‘id-as-service’ more trustworthy? Consider the following. A client presentsa key, and gets one or more values back. An example key is a biometric (e.g. pass phrase,fingerprint, iris, face, palm, etc) plus a possible additional parameter (e.g. age verify, bankaccount number). The response value is returned: “is over 21”, bank a/c, “is entitled toNHS care” etc. In some system designs, what is returned is a token (or collection of tokens)that have a sole purpose of authentication and are of no use or meaning to anyone else.The client side should run with security, up to and possibly including client user context,such as knowing who can see the display or know the location etc. The network shouldat least implement basic security such as Transport Layer Security (TLS). The server sideshould ensure and keep all data encrypted. It is possible to run the server in an enclave (SGX,Trustzone etc). However, a problem that can occur is if these are subsequently compromised,but we continue to use anyway (i.e. confidential cloud) with relatively low performancepenalty. Enclaves also potentially provide attestation (e.g. of integrity), which can also beuseful but might depend on a single authority that has to be trusted and trustworthy. Wecould run the server key/value lookup using FHE. In this case the problem is performance,look up rate could have a pretty low throughput. However, see this service which claimsotherwise:- http://privatebiometrics.com/index.html .We could run the server with data sliced or sharded (that is disaggregated), and use MPCto do match key to value. This has some latency challenges, but is not computationallypathological, and scales out well. Note others have built solutions to privacy in this spacetoo, e.g. https://cryptpad.fr/We could distribute data over many cloud services and federate. Alternatively, it is possibleto run a fully distributed bespoke system (possibly non virtualized/not cloud. The simplestwould be to put the key/value store on a P2P Chord/Distributed Hash Table like Kademlia.This basically mimics regular password file (hash onto a file, but in this case, has onto anode in Kademlia). Kademlia also supports resilience/node failure recovery and has highperformance. There is no access control inherently, but it could be added.Another candidate for this is a distributed ledger system (DLT), such as Ethereum orHyperledger, with one added benefit that this has high integrity, and is effectively tamperproof. Ledgers can be fully peer to peer (p2p), and therefore permission-less, or dependupon an access control system that itself could be distributed or centralized (permissioned).Mutable data has to be kept off chain, or some new construct applied. DLT also supportcomputation, as part of transactions, and IBM hase proposed adding MPC as part of thesecomputations.There is a slight circularity here, in that the sign on for a permissioned system itself requires eyond COVID-19: Network science and sustainable exit strategies
Systems operators or customers may wish to carry out statistical analyses to audit the properoperations of identity systems. These operations need not be privileged. Differential Privacyis one mechanism to provide privacy with respect to an individual’s data in a set. i.e. ifsomeone is doing queries that return aggregates, this only returns results as if the individualrecord was not there.Applications that use id might also need analytics, for example public health researcherswant to look at contact tracing statistics to determine infection rates between users anddifferent classes of user (age, gender etc).So this might be useful here, but note this has to be for authorized users only (role basedaccess control may be needed), and there is a limit/quota on number of queries, that is“budget” must be traded off against precision.Decentralized systems can be coupled with randomness to provide prevention of trawling,with accurate lookups provided by the model run on users own device and data. But seeabove for risk problem with decentralized.For different organizations and nations, trust assumptions may vary - eg. do you trust ahealth service provider, government, a bank, a set of individuals, an infrastructure, hardware,operating systems etc Depending on the answer, a different mix of choices from the menuabove may be appropriate.
Here, we provide a brief summary of motivations, concerns and proposals relating toimmunity passports. We do not intend this to provide the reader with a complete overview,only to inform them of some of the basics. eyond COVID-19: Network science and sustainable exit strategies eyond COVID-19: Network science and sustainable exit strategies
7. Measuring efficacy and impact of COVID-19 mitigation methods
Given the spread of the ongoing SARS-CoV-2 pandemic, automated contact tracing has beensuggested as an effective means of containing the spread of the virus while enabling a societyto reopen its economy safely. Consequently, a more detailed and rigorous examination ofthe efficacy of automated contact tracing is required given the distinct difference in theprevalence of this pandemic from the ones in the recent past and the different modes oftransmission of the pathogen.Manual contact tracing, used more traditionally, has been observed to be effective in previousepidemics caused by the Ebola virus, SARS-CoV and MERS-CoV [94, 95, 96, 97, 98]. Manualcontact tracing is not very effective against pathogens that spread like the influenza virus butis more effective for containing smallpox and SARS-CoV [99]. The viral shedding patternsof SARS-CoV and MERS-CoV are similar [100, 101] and show almost no pre-symptomatictransmission [51], while Ebola is known to be transmitted through the bodily fluids ofinfected individuals after the onset of symptoms [102]. On the other hand, influenza shows asignificant rate of viral shedding in the pre-symptomatic stage [103]. The spreading patternof SARS-CoV-2 is similar to influenza and quite different from Ebola or SARS-CoV. eyond COVID-19: Network science and sustainable exit strategies • Let N be the number of individuals in a population and f i the fraction of the populationthat is infected, regardless of whether they know it or not. Therefore, the true numberof infected individuals is f i N . • If testing is conducted only when mild or severe symptoms are seen (i.e. excludingtesting of asymptomatic cases), the number of confirmed cases is r c f i N with r c beingthe fraction of the infected that will be confirmed as infected by testing. • We define f e as the fraction of the population that is enrolled for automated contacttracing and f c as the fraction of the users that will confirm that they have beendiagnosed positive. Hence, the number of individuals that have tested positive, areusing automated contact tracing and will confirm that they are sick is f c f e r c f i N . • We define a c as the average number of contacts per person in the period of time t whoare at risk of being infected due to proximity with a sick individual and is assumed tobe greater than 0.Using these quantities, we can estimate the number of individuals that can be traced as f c f e r c f i N × a c × f e , i.e. (the number of reported positive tests) × (the fraction of contactsthat will be notified per the report). For automated contact tracing to work effectively,this number should be greater than or equal to the number of individuals that need to bequarantined or isolated since they are now at risk of being infected from coming in contactwith a sick person. For the evaluation, we define the following. • Since p t is defined as the probability of transmission of infection within the proximityradius r being exposed for a time greater than t , the number of individuals at risk is,at most p t f i N a c , i.e., p t × (number of contacts of the group of infected individuals). • Finally, we define f T as the fraction of the individuals at risk of being infected thatneeds to be successfully quarantined to quell the spread of the pathogen.Therefore, the number of individuals that should be quarantined is f T p t f i N a c . Forautomated contact tracing to slow down the spread of the virus effectively, we have, f e f c r c f i N a c ≥ f T p t f i N a c . (15) eyond COVID-19: Network science and sustainable exit strategies a c , the average number of contacts, drops out of the inequality. Hence, theinequality is independent of the population density of the region. This is because eq. (15)is in terms of fraction of the population and not the absolute number of individuals. Thissimply implies that in a region of denser population a larger number of people need to becontacted and quarantined but leaves f e independent of the population density. Since theright-hand side is the minimum fraction of the population that needs to be traced we arriveat: f mine = (cid:115) f T p t f c r c . (16)The fraction f mine is the minimum fraction of the population that needs to be enrolled inautomated contact tracing for it to be effective as a means of slowing down the spread ofthe pandemic.Let us examine the limit p t = f c = r c = 1. This is the limit where every significant contactis assumed to be at risk, everyone who is enrolled in the automated contact tracing programreports sick when tested positive and every sick individual can be successfully identified bytesting. Then we arrive at the relation f mine = √ f T . Since f T is the fraction of contacts thatneed to be successfully isolated, it can be extracted from the abscissa of Fig. 3 of ref. [50].For example, if 100% of the confirmed infected cases can be isolated, then for a change in theepidemic growth rate by − .
1, one needs f T ∼
60% and hence f mine ∼ f mine scales as the square root of f T since both the infected and the contact at risk needto be enrolled and the probability that each are enrolled is f e leading to f T ∝ f e . It givesthe threshold which f mine cannot exceed for any given f T . Several scenarios of the parametersets are studied in ref. [104]. The necessary scale of implementation of automated contacttracing is too large for it to be considered by itself an effective measure to slow down theongoing pandemic. For automated contact tracing to be a viable option, f mine has to be aslow as possible. A closer look at the parameters reveals the following: • Both f T and p t depend on the dynamics of the disease spread. The fraction of tracedcases that need to be quarantined to stop the spread of the disease, f T , can be reducedby extensive monitoring of the disease to make sure sick cases are isolated as soon aspossible and their contacts are traced. Even a day or two of delays can increase f T making automated contact tracing ineffective [50]. • Variations in p t can be caused by several factors some of which are controllable. Since p t depends on the contagiousness of the disease and any protective measures taken againstthe spread of the infection, p t can be reduced by measures of limited social distancing,the use of PPE and raising public awareness about the contagiousness of COVID-19.This can pose a significant challenge in densely populated regions and regions with poor eyond COVID-19: Network science and sustainable exit strategies • f c is somewhat more difficult to control assuming the reporting of those who areconfirmed sick is voluntary. This can only be increased by increasing the population’swillingness to contribute to automated contact tracing. • r c is the parameter that is least under control since without very large-scale testing,asymptomatic and mildly symptomatic cases will be difficult to find. This is especiallytrue if the infection can spread by means other than proximity alone as might be thecase for SARS-CoV-2 [105, 106, 107].Thus we see that a combination of several measures along with a large participation ofthe population in contact tracing would be the optimal solution for avoiding extensivepopulation-wide social distancing measures and reducing the cost to the economy and well-being of a nation and also allow for greater freedom of movement during a pandemic. Inthe following section, we discuss future studies on other possible measures that can helpin the mitigation of the spread of COVID-19 with particular focus on computational andalgorithmic approach leveraging data science and network theory. A pandemic is a population-wide crisis, yet it affects individuals at varying degrees ofcriticality depending not only on their personal lifestyles but also on the local demographics.It has a unique way of amplifying what history has created as a lingering effect on the currentsocio-economic trajectories of the immediate neighborhood that any individual lives in. Inessence, the effects of and strategies during a pandemic cannot be disentangled from the localsocio-economic conditions and historical fluctuations. This renders any large-scale policyimplementation without considering the local conditions at the county, state or national levelcompletely ineffective during a pandemic even though the response to it must be population-wide to sufficiently mitigate it. What is imperative looking forward is a means of analyzinglocalized datasets to understand the curation of mitigation procedures during the onset of awave of disease spreading. Firstly, data needs to be collected with sufficient granularity toallow for inferences on local prevalence. Secondly, the spread of the disease has to be studiedvis--vis the local demographics of any neighborhood. Considering the fact that these datasetswill be highly multivariate and have very complex correlation pattern obfuscating the causalconnections between the driver and the driven, advanced modeling methods and statisticaltools are imperative for the understanding of disease spread at a microscopic level of thesocio-economic structure of a nation. We have undertaken a three-fold study of variousaspects of disease spread and containment that will link them to exit strategies throughthe mathematical models of curated socio-economic responses and the study of immunity eyond COVID-19: Network science and sustainable exit strategies
Left: distribution of population density in the USA. Right: COVID-19 diseaseprevalence per 100,000 individuals in each county. The COVID-19 prevalence distributionis quite different from the population density distribution. Plots made with the HighchartsMaps JavaScript library from Highcharts.com with a CC BY-NC 3.0 license. development in a community. These three parts of the study have a deep underlying linkwhich adds to the strength of the analysis that we propose. The three parts can be describedas:(i) A study of the correlations between several socio-economic metrics and geospatialdemographics at the county level and their correlation with COVID-19 prevalence.(ii) A study of immunity development in a community based on a detailed agent-basedmodel and the training of a machine learning algorithm to probabilistically assess andcategorize the immunity development in individuals.(iii) A network-based model of urban areas to understand curated closures of the commercialand industrial sectors to find an optimal level between uncontrolled disease spread anddamages to the economy.All three topics tie into a common goal: understanding the socio-economic conditions thataffect the spreading of the pandemic and devising effective exit strategies and mitigatingpolicies that can allow for the sustenance of economy while allowing for lower footprintof a pandemic in terms of human lives lost and perturbations to the social norm. Thesestrategies will be augmented by an understanding of immunity development which is crucialfor increasing mobility within a population and will build tools that will allow for algorithmicassessments to aid in prioritizing clinical testing for immunity. In what follows, we give somedetails of the ideas that we are exploring in these studies.
As COVID-19 spreads by contact andproximity, it is natural to assume that the pandemic will have its worst effects in regions eyond COVID-19: Network science and sustainable exit strategies
In another work we study thedisease spread using an agent-based model [57, 58] and develop a machine learning algorithmthat will be able to identify immune individuals in a data-driven manner. The study requiresa simulation of an agent-based model to create a simulated data-set of immune agents startingwith a small number of sick agents. To set up the rules for the agent-based model there hasto be an understanding of how the disease propagates in real communities affected by thedisease. This requires the gathering knowledge from emerging clinical studies of COVID-19affected communities. The immunity detection algorithm is trained using the simulated data-set generated by the agent-based model. This algorithm will evolve into a semi-supervisedmachine learning algorithm that will learn the optimal values of the parameters necessary forinferring on immunity development in individuals. Data obtained in 7.2.1, related to localinfection prevalence, duration of pre-symptomatic and symptomatic stages of the infection,demographic data on infected individuals, assessment of the prevalence of asymptomaticcases helps in the design of the simulation which will be used to train the immunity detectionalgorithm.From this we will gain a data-driven understanding of immunity development in a population.This will act as a starting point when real data is made available with the deployment ofanti-body tests. An assessment of individual immunity based on prior exposure, underlyinghealth conditions, local disease spread data and mobility history will allow individuals to beaware of possible immunity development and allow any institution (within the healthcaresystem or otherwise) to prioritize immunity tests based on algorithmic assessment. This willallow for better assessment of the necessity for quarantining individuals and will ease therequirement for population-wide stay-at-home orders, and can ultimately be a reference tomaking policy decision for reopening for COVID-19 as well as future epidemics. eyond COVID-19: Network science and sustainable exit strategies Exit strategies duringan ebbing pandemic requires special caution so as to not trigger its resurrection as is beingseen in several parts of the world now. A key component is understanding the fraction ofthe various commercial sectors that can be opened to sustain the flow of the economy whileoptimizing the social contact within the population. There are certain businesses that havea higher footfall and hence act as transmission hubs like restaurants and supermarkets. If afraction of these hubs is closed intermittently, it can reduce the effective pathway for diseasepropagation, hence, slowing down the disease spread while it allows keeping the businessesat a sustainable level. However, this requires accurate large-scale prediction about whatfraction of different business sectors need to be closed. To this end, we are developingmulti-layered network models [108, 109, 110] to represent businesses as nodes with theirinterdependence as links in one layer and the pathways of disease spread in another layer.Utilizing the business interaction network on the first layer, we investigate how the partialclosure of certain businesses affect other businesses that depend on them. Then, we analyzemobility patterns on the business network by using mobility data collected from Google andidentify hubs associated with high mobility in an urban area. We adopt the agent-basedmodel discussed in 7.2.2 to simulate the mobility data and tune it with the fraction of openbusiness on the first layer to reduce the mobility at the hubs. This enables the analysis ofhow the intervention on the business network propagates and influence the spread of diseaseacross the social interaction network on the second layer [111, 112].
8. Concluding Remarks
Since the workshop, the pandemic has significantly worsened, and the policy responses toupsurges in cases has varied widely from nation to nation, and even state to state. As ofat September 27 th , the global cases are approaching 33 million, and sadly deaths now standat 995 ,
583 with a difficult Northern Hemisphere winter to come. The impact of restrictionson personal liberty and economic activity continue to be manifested in deep recessions,significant changes in patterns of travel, and increasingly in civil unrest. The importanceof seeking smart solutions to manage the pandemic with minimal disruption is even morepressing than when the workshop was convened. We hope that the results presented herewill inform both researchers and policy makers.It is clear that there is much left to be understood regarding the interplay between thenetwork of contacts and the progression of the epidemic. As was explained in Sections 2-5 the precise structure of the contact network. However, detailed information about thenetwork structure is scarce and most of the modeling approaches have been based on coarse-grained data about the network structure. An exception to this rule is the case in whichdata from track and tracing apps is taken into account, but even here, the data includes onlypartial information and needs to satisfy the privacy agreement set up with its customers. eyond COVID-19: Network science and sustainable exit strategies
Acknowledgments
I.Z. Kiss and F. Di Lauro acknowledge support from the Leverhulme Trust for the ResearchProject Grant RPG2017-370. eyond COVID-19: Network science and sustainable exit strategies References [1] Y.-C. Wu, C.-S. Chen, and Y.-J. Chan, “The outbreak of covid-19: An overview,”
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